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Sagot :
Bien, vamos a realizar las operaciones pedidas para cada caso utilizando los valores dados de \( z \) y \( w \). Vamos a calcular tanto el producto \( zw \) como la división \( \frac{z}{w} \).
### (a) \( z = -5 + 4i \), \( w = 2 - 3i \)
Producto \( zw \):
[tex]\[ (-5 + 4i) \cdot (2 - 3i) = (-5) \cdot 2 + (-5) \cdot (-3i) + (4i) \cdot 2 + (4i) \cdot (-3i) \][/tex]
[tex]\[ = -10 + 15i + 8i - 12i^2 = -10 + 23i + 12 \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = 2 + 23i \][/tex]
División \( \frac{z}{w} \):
Para realizar la división de números complejos, multiplicamos numerador y denominador por el conjugado del denominador:
[tex]\[ \frac{-5 + 4i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} = \frac{(-5)(2) + (-5)(3i) + (4i)(2) + (4i)(3i)}{(2)^2 + (3i)^2} \][/tex]
[tex]\[ = \frac{-10 - 15i + 8i + 12}{4 + 9} = \frac{-10 - 7i + 12}{13} \][/tex]
[tex]\[ = \frac{2 - 7i}{13} = -1.6923 - 0.5385i \][/tex]
### (c) \( z = -3 - 2i \), \( w = -5 + i \)
Producto \( zw \):
[tex]\[ (-3 - 2i) \cdot (-5 + i) = (-3)(-5) + (-3)(i) + (-2i)(-5) + (-2i)(i) \][/tex]
[tex]\[ = 15 - 3i + 10i - 2i^2 = 15 + 7i + 2 \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = 17 + 7i \][/tex]
División \( \frac{z}{w} \):
[tex]\[ \frac{-3 - 2i}{-5 + i} \cdot \frac{-5 - i}{-5 - i} = \frac{15 + 3i - 10i - 2i^2}{25 + 1} = \frac{15 - 7i + 2}{26} \][/tex]
[tex]\[ = \frac{17 - 7i}{26} = 0.5 + 0.5i \][/tex]
### (e) \( z = 5 - 2i \), \( w = 6i \)
Producto \( zw \):
[tex]\[ (5 - 2i) \cdot 6i = (5 \cdot 6i) + (-2i \cdot 6i) = 30i - 12i^2 \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = 30i + 12 = 12 + 30i \][/tex]
División \( \frac{z}{w} \):
[tex]\[ \frac{5 - 2i}{6i} = \frac{5 - 2i}{6i} \cdot \frac{-i}{-i} = \frac{(5 \cdot -i - 2i \cdot -i)}{(6i \cdot -i)} = \frac{-5i + 2i^2}{-6} \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = \frac{-5i - 2}{-6} = \frac{2 + 5i}{6} = -0.3333 - 0.8333i \][/tex]
### (g) \( z = -9 + 7i \), \( w = 4 + 9i \)
Producto \( zw \):
[tex]\[ (-9 + 7i) \cdot (4 + 9i) = (-9)(4) + (-9)(9i) + (7i)(4) + (7i)(9i) \][/tex]
[tex]\[ = -36 - 81i + 28i + 63i^2 = -36 - 53i + 63 \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = -99 - 53i \][/tex]
División \( \frac{z}{w} \):
[tex]\[ \frac{-9 + 7i}{4 + 9i} \cdot \frac{4 - 9i}{4 - 9i} = \frac{(-9)(4) + (-9)(-9i) + (7i)(4) + (7i)(-9i)}{(4)^2 + (9i)^2} = \frac{-36 + 81i + 28i - 63i^2}{16 + 81} \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = \frac{-36 + 109i + 63}{97} = \frac{27 + 109i}{97} = 0.2784 + 1.1237i \][/tex]
Resumimos los resultados para cada caso:
(a) \( zw = 2 + 23i \), \(\frac{z}{w} = -1.6923 - 0.5385i \)
(c) \( zw = 17 + 7i \), \(\frac{z}{w} = 0.5 + 0.5i \)
(e) \( zw = 12 + 30i \), \(\frac{z}{w} = -0.3333 - 0.8333i \)
(g) [tex]\( zw = -99 - 53i \)[/tex], [tex]\(\frac{z}{w} = 0.2784 + 1.1237i \)[/tex]
### (a) \( z = -5 + 4i \), \( w = 2 - 3i \)
Producto \( zw \):
[tex]\[ (-5 + 4i) \cdot (2 - 3i) = (-5) \cdot 2 + (-5) \cdot (-3i) + (4i) \cdot 2 + (4i) \cdot (-3i) \][/tex]
[tex]\[ = -10 + 15i + 8i - 12i^2 = -10 + 23i + 12 \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = 2 + 23i \][/tex]
División \( \frac{z}{w} \):
Para realizar la división de números complejos, multiplicamos numerador y denominador por el conjugado del denominador:
[tex]\[ \frac{-5 + 4i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} = \frac{(-5)(2) + (-5)(3i) + (4i)(2) + (4i)(3i)}{(2)^2 + (3i)^2} \][/tex]
[tex]\[ = \frac{-10 - 15i + 8i + 12}{4 + 9} = \frac{-10 - 7i + 12}{13} \][/tex]
[tex]\[ = \frac{2 - 7i}{13} = -1.6923 - 0.5385i \][/tex]
### (c) \( z = -3 - 2i \), \( w = -5 + i \)
Producto \( zw \):
[tex]\[ (-3 - 2i) \cdot (-5 + i) = (-3)(-5) + (-3)(i) + (-2i)(-5) + (-2i)(i) \][/tex]
[tex]\[ = 15 - 3i + 10i - 2i^2 = 15 + 7i + 2 \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = 17 + 7i \][/tex]
División \( \frac{z}{w} \):
[tex]\[ \frac{-3 - 2i}{-5 + i} \cdot \frac{-5 - i}{-5 - i} = \frac{15 + 3i - 10i - 2i^2}{25 + 1} = \frac{15 - 7i + 2}{26} \][/tex]
[tex]\[ = \frac{17 - 7i}{26} = 0.5 + 0.5i \][/tex]
### (e) \( z = 5 - 2i \), \( w = 6i \)
Producto \( zw \):
[tex]\[ (5 - 2i) \cdot 6i = (5 \cdot 6i) + (-2i \cdot 6i) = 30i - 12i^2 \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = 30i + 12 = 12 + 30i \][/tex]
División \( \frac{z}{w} \):
[tex]\[ \frac{5 - 2i}{6i} = \frac{5 - 2i}{6i} \cdot \frac{-i}{-i} = \frac{(5 \cdot -i - 2i \cdot -i)}{(6i \cdot -i)} = \frac{-5i + 2i^2}{-6} \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = \frac{-5i - 2}{-6} = \frac{2 + 5i}{6} = -0.3333 - 0.8333i \][/tex]
### (g) \( z = -9 + 7i \), \( w = 4 + 9i \)
Producto \( zw \):
[tex]\[ (-9 + 7i) \cdot (4 + 9i) = (-9)(4) + (-9)(9i) + (7i)(4) + (7i)(9i) \][/tex]
[tex]\[ = -36 - 81i + 28i + 63i^2 = -36 - 53i + 63 \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = -99 - 53i \][/tex]
División \( \frac{z}{w} \):
[tex]\[ \frac{-9 + 7i}{4 + 9i} \cdot \frac{4 - 9i}{4 - 9i} = \frac{(-9)(4) + (-9)(-9i) + (7i)(4) + (7i)(-9i)}{(4)^2 + (9i)^2} = \frac{-36 + 81i + 28i - 63i^2}{16 + 81} \][/tex]
Recordando que \( i^2 = -1 \):
[tex]\[ = \frac{-36 + 109i + 63}{97} = \frac{27 + 109i}{97} = 0.2784 + 1.1237i \][/tex]
Resumimos los resultados para cada caso:
(a) \( zw = 2 + 23i \), \(\frac{z}{w} = -1.6923 - 0.5385i \)
(c) \( zw = 17 + 7i \), \(\frac{z}{w} = 0.5 + 0.5i \)
(e) \( zw = 12 + 30i \), \(\frac{z}{w} = -0.3333 - 0.8333i \)
(g) [tex]\( zw = -99 - 53i \)[/tex], [tex]\(\frac{z}{w} = 0.2784 + 1.1237i \)[/tex]
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